COCOMPACT LATTICES ON Ã_n BUILDINGS

We construct cocompact lattices Γ′₀ < Γ₀ in the group G = PGL_d (double-struck F_q((t))) which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright-Steger Isr. J. Math. 103 (1998), 125-140. The stabiliser of each vertex in Γ′₀ is a Singer cycle and the stabiliser of each vertex in Γ₀ is isomorphic to the normaliser of a Singer cycle in PGL_d (q). We show that the intersections of Γ′₀ and Γ₀ with PSL_d (double-struck F_q((t))) are lattices in PSL_d(double-struck F_q((t))), and identify the pairs (d, q) such that the entire lattice Γ′₀ or Γ₀ is contained in PSL_d(double-struck F_q((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL₃(double-struck F_q((t))). Our proofs combine the construction of Cartwright-Steger Isr. J. Math. 103 (1998), 125-140 with results about Singer cycles and their normalisers, and geometric arguments.